and \(0 \le \alpha , \beta \le 1\)  such that
\(\beta >\max \lbrace 1-2 \alpha , \frac{1}{2}(1-\alpha ) \rbrace \)  guarantee that the energy of weak solutions is locally conserved.
Very recently, the hypothesis \(\pi \in C^{2}\)  was improved to \(\pi \in C^{1,\gamma -1}\)  with \(1\le \gamma <2\)  by Akramov-Debiec-Skipper-Wiedemann in [1]}. Moreover, under the density \(\varrho \in L^{\infty } ([0, T] \times \mathbb {T}^{d} ) \) ,
sufficient conditions for weak solutions of system (REF ) and (REF ) to conserve the energy can be found in [2]}, [3]}, [4]}, [5]}.
